How to complete this proof process of Arzelà-Ascoli theorem?

Let's prove that $F$ is totally bounded

Take $\epsilon \gt 0$. As $\mathcal F$ is supposed to be equicontinuous, for all $x \in X$, it exists an open neighborhood $\mathcal O_x$ such that $d(f(x),f(y)) \lt \epsilon/4$ for all $ f \in \mathcal F$ and all $y \in \mathcal O_x$.

As $X$ is compact, we can extract from $\{\mathcal O_x \mid x \in X\}$ a finite cover $\mathcal O_{x_1}, \dots \mathcal O_{x_m}$.

For all $i \in \{1, \dots, m\}$ $\mathcal F(x_i)$ is precompact and therefore so is their union $\mathcal U$ (a finite union of precompact subsets is precompact). Let $\{u_1, \dots, u_n\}$ be a finite subset of $\mathcal U$ such that $\mathcal U \subseteq \bigcup_{j=1}^n B(u_j, \epsilon/4)$ where $B(x, r)$ stands for the open ball centered on $x$ of radius equal to $r$.

Now take $f(x) \in F$. It exists $x_i$ such that $x \in \mathcal O_{x_i}$ and $u_j$ such that $f(x_i) \in B(u_j, \epsilon/4)$. Therefore

$$d(f(x),u_j) \le \underbrace{d(f(x),f(x_i))}_{x \in \mathcal O_{x_i}} + \underbrace{d(f(x_i),u_j)}_{f(x_i) \in B(u_j, \epsilon/4)} \le \epsilon/4 + \epsilon/4 \le \epsilon/2$$ which proves that $$F \subseteq \bigcup_{j=1}^n B(u_j, \epsilon/2)$$

As the diameter of each of the $B(u_j, \epsilon/2)$ is less or equal to $\epsilon$, we can conclude that $\mathcal F$ is totally bounded and to the desired result.